Probability distribution of an integral quadratic functional on the trajectories of a complex-valued Ornstein-Uhlenbeck process

Virchenko, Yu. P.; Mazmanishvili, A. S.

doi:10.1007/s10559-005-0029-4

Cited by 4 publications

(5 citation statements)

References 7 publications

(7 reference statements)

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“…Он позволяет сводить нахождение математических ожиданий для более сложных квадратичных функционалов от траекторий процесса (см. [24][25][26][27]). В настоящей работе мы следуем методу, который был применен ранее одним из авторов статьи в [28], [29].…”

Section: 𝑑 𝑑𝑡unclassified

See 1 more Smart Citation

The Ziegert formula for multidimensional Ornshtein-Uhlenbeck random processes

Virchenko

Vitokhina

2021

Прикладная Математика &Amp; Физика

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Section: 𝑑 𝑑𝑡unclassified

“…[24][25][26][27]). В настоящей работе мы следуем методу, который был применен ранее одним из авторов статьи в [28], [29].…”

Section: 𝑑 𝑑𝑡unclassified

The Ziegert formula for multidimensional Ornshtein-Uhlenbeck random processes

Virchenko

Vitokhina

2021

Прикладная Математика &Amp; Физика

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“…However, in practical computation we cannot analytically separate the complex-valued wave function into two real wave functions as described by Equation (32), since they are coupled by the complex Schrödinger equation. For such a long time, physicists do not know why the Schrödinger equation is complex, including Schrödinger himself.…”

Section: Complex Random Motion In Complex Mechanicsmentioning

confidence: 99%

Extending Quantum Probability from Real Axis to Complex Plane

Yang¹,

Han²

2020

Preprint

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Probability is an open question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the domain of probability extends to the complex space, including the generation of complex trajectories, the definition of the complex probability, the relation of the complex probability to the real quantum probability, and so on. The complex treatment proposed here applies the optimal quantum guidance law to derive the stochastic differential (SD) equation governing the particle’s random motions in the complex plane. The ensemble of the complex quantum random trajectories (CQRTs) solved from the complex SD equation is used to construct the probability distribution ρc(t,x,y) of the particle’s position over the complex plane z=x+iy. The correctness of the obtained complex probability is confirmed by the solution of the complex Fokker-Planck equation. The significant contribution of the complex probability is that it can be used to reconstruct both quantum probability and classical probability, and to clarify their relationship. Although quantum probability and classical probability are both defined on the real axis, they are obtained by projecting complex probability onto the real axis in different ways. This difference explains why the quantum probability cannot exactly converge to the classical probability when the quantum number is large.

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“…However, there exist simple analytical approximations that give a good overview of the behavior of such functionals. For example Mazmanishvili (2000), Virchenko and Mazmanishvili (2004) give the following form:…”

Section: R)f (I ( R) E( R))mentioning

confidence: 99%

A comprehensive study of the most likely path formalism for proton-computed tomography

Erdélyi

2009

Phys. Med. Biol.

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We investigate some generalizations of the most likely path formalism developed for proton-computed tomography. The stochastic path of a proton inside a homogeneous medium is replaced by a deterministic smooth path that maximizes the probability of the proton passing through the points on this curve, given measured entrance and exit parameters for each individual proton. We study various factors that influence this curve and the associated error envelopes. These factors are the influence of the energy loss, a logarithmic correction factor in the small angle Coulomb scattering and the importance of path length versus material thickness. We develop a method for further constraining the possible proton paths by including energy information in the derivation of the most likely path, utilizing an infinite-dimensional constrained functional analysis method. It is shown that while there is an additional uncertainty on the most likely path that is difficult to determine experimentally, the associated error envelopes are smaller, resulting in possibly slightly improved spatial resolution for proton-computed tomography.

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Probability distribution of an integral quadratic functional on the trajectories of a complex-valued Ornstein-Uhlenbeck process

Cited by 4 publications

References 7 publications

The Ziegert formula for multidimensional Ornshtein-Uhlenbeck random processes

The Ziegert formula for multidimensional Ornshtein-Uhlenbeck random processes

Extending Quantum Probability from Real Axis to Complex Plane

A comprehensive study of the most likely path formalism for proton-computed tomography

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